On complex reflection groups G(m, 1, r) and their Hecke algebras

Abstract

We construct an algorithm for getting a reduced expression for any element in a complex reflection group G(m, 1, r) by sorting the element, which is in the form of a sequence of complex numbers, to the identity. Thus, the algorithm provides us a set of reduced expressions, one for each element. We establish a one-one correspondence between the set of all reduced expressions for an element and a set of certain sorting sequences which turn the element to the identity. In particular, this provides us with a combinatorial method to check whether an expression is reduced. We also prove analogues of the exchange condition and the strong exchange condition for elements in a G(m, 1, r). A Bruhat order on the groups is also defined and investigated. We generalize the Geck-Pfeiffer reducibility theorem for finite Coxeter groups to the groups G(m, 1, r). Based on this, we prove that a character value of any element in an Ariki-Koike algebra (the Hecke algebra of a G(m, 1, r)) can be determined by the character values of some special elements in the algebra. These special elements correspond to the reduced expressions, which are constructed by the algorithm, for some special conjugacy class representatives of minimal length, one in each class. Quasi-parabolic subgroups are introduced for investigating representations of Ariki- Koike algebras. We use n x n arrays of non-negative integer sequences to characterize double cosets of quasi-parabolic subgroups. We define an analogue of permutation modules, for Ariki-Koike algebras, corresponding to certain subgroups indexed by multicompositions. These subgroups are naturally corresponding, not necessarily one-one, to quasi-parabolic subgroups. We prove that each of these modules is free and has a basis indexed by right cosets of the corresponding quasi-parabolic subgroup. We also construct Murphy type bases, Specht series for these modules, and establish a Young's rule in this case.